The Social Satisfaction:

a Fairness Theory about Income Distribution

with Applications in China

Ouyang KuiInstitute of Quantitative & Technical Economics,

Chinese Academy of Social Sciences

1 Introduction

• Economic development, income growth and social welfare• The Dalton-Atkinson’s approach (Dalton, 1920; Atkinson, 1970) • The choice of SWFs and the choice of utility functions• The dictatorship conclusion(d’Aspremont & Gevers, 1977) and

Arrow’s impossibility theorem (Arrow, 1963)• The Nash SWF(Nash, 1950)• Revealed preferences and subjective satisfaction: Ordinalism vs

Cardinalism (Mandler,2006)• The axiomatic characterization of the measure of income

distribution

2 The Nash SWF: A differential equation approach

• Definition 2.1 A SWF is homogeneous of degree k:

• Definition 2.2 A SWF is symmetrically differentiable:

duuW

duuW

duuW

dWN

2

21

1

duuW

duuW

duuW

N

2

21

1

),,(),,( 11 uuWttutuW kN

• Theorem 2.3 The only homogeneous, symmetrically differentiable SWF is the linear power transformation of Nash SWF:

0,),,,(0

21

CuCuuuW

Nk

N

iiN

3 The Social Satisfaction

• 3.1 The SF: a fuzzy measure of utility Definition 3.1 The individual satisfaction function (SF):

S: RN→[0,1].• 3.2 The SSF: a normative on SWF Definition 3.2.1 The SWF W(S1, …, SN) is a social satisfaction

function (SSF) if we have

Theorem 3.2.2 The unique homogeneous and symmetrically differentiable SSF is the geometric average of individual SFs.

NjiSSWSSji jiji ,,2,1,,,,

• 3.3 The invariance properties of SWF

),,(

)(

),,(

),,(),,(

''11

1

'

1111

111

''1

1

'

11

NNNN

N

iii

N

ii

N

iii

N

ii

N

iiiii

NNN

N

N

iii

N

iiiN

uuW

dufdufudf

uuW

uuWdufdufuuW

4 Fairness and equality in income distribution

• 4.1 The Nash bargaining problem:

The Nash solution to the bargaining problem (impartiality):

j

j

ji

i

i x

S

SxS

S

11

0)(..

)(max

1

1

N

ii

NN

iii

xGts

xSW

• 4.2 The general form of satisfaction function:

• 4.3 The social satisfaction index (SSI) of income distribution

The Nash solution:),exp()(i

iii x

rxS )

1exp(

1

N

i i

i

xr

NW

jj

ii x

rr

x

))(exp()(0x

dttfCxS

4.4 Be equal of welfare or income?

Inequality in different solutions• The Egalitarian SSF: The Egalitarian solution:• The Nash SSF: The Nash solution: • The Utilitarian SSF: The Utilitarian solution:

ji SS

jxjixi SS11

jxjS

jixiS

i SS

),min( 21 SSW

21SSW

21 21

21

SSW

Inequality in different solutions(r1=1, r2=1.2) (r1=1, r2=1.6) (r1=1, r2=2) (r1=1, r2=4)

x S W x S W x S W x S W

E

4.54555.4545

.8025

.8025 .8025

3.84626.1538

.7711

.7711 .7711

3.33336.6667

.7408

.7408 .7410

2.00008.0000

.6065

.6065 .6065

N

4.77235.2277

.8110

.7949 .8029

4.41525.5848

.7973

.7509 .7738

4.14215.8579

.7855

.7108 .7472

3.33336.6667

.7408

.5488 .6376

U

4.80035.1996

.8119

.7939 .8029

4.50005.5000

.8007

.7476 .7742

4.28415.7159

.7918

.7048 .7483

3.76286.2372

.7666

.5266 .6466

5 The axiomatic characterization of the SSI of income distribution

• Theorem 5.1 If for all x R, the SF is second-∈order differentiable, S(0) = 0, S(+∞) = 1, then we have

• Definition 5.2 A SF has logarithmic constant elasticity if for all we have

)('

)(''

)(')(''

)(')(')()(xS

xS

xSxS

xSxSxSxSj

j

i

ijiji

0,

)(')( rr

xInSxInSx

i

i

• Theorem 5.3 A SF has logarithmic constant elasticity if and only if it can be generated from

If all SFs have logarithmic constant elasticity, then the SSI can be expressed as:

• Property 5.4 (Transfers principle) Y is obtained from X and for some i and j, (a)Si(xi)<Sj(xj); (b) xi-yi=yj-xj>0; (c) xk=yk for all k≠i,j, we have: (1)If xi<xj, then W(X) > W(Y); (2)More generally, if yi<yj, then W(X) > W(Y).

N

ii

i

iN xNxxW

11 .0,0),

1exp(),,(

NixxS iiii ,,2,1,0,0),exp()(

• Property 5.5 (Independent of income units) If W(Y) = W(X), then for t > 0, W(tY) = W(tX).

• Property 5.6 (Replication principle) If a society N(Y, Sn×m) is a replication of another society M(X, Sm), Y=Xn, X=xm, then W(X) = W(Y).

• Property 5.7 (Geometric Decomposability) In a society of N agents, for N = n + m, then

Generally, for N =n1+ …+nm, then we have

mn

mnnn

NN WWW 1

1

mm

nn

NN WWW

• If we set for all i, i.e. the SSI is symmetric, then the SSI can be defined as:

• Property 5.8 (Symmetry) If Y is obtained from X by a permutation of incomes, then W(Y) = W(X).

• Property 5.9 (Pigou-Dalton transfers principle) If is obtained from such that for some i and j, (a)xi<xj; (b) xi-yi=yj-xj>0; (c) xk=yk for all k≠i,j, then W(Y) > W(X).

N

iiN x

NxxW

11 .0,0),exp(),,(

• Property 5.10 (Population principle) If Y is a replication of X, then W(X) = W(Y).

• Property 5.11 (Homogeneity) If Y = tX, t>0, then W(Y) = W(X) if we set

• Theorem 5.12 Let W(xi)=Si(xi). Then the unique index W of income distribution satisfies the geometric decomposability (Property 5.7) for all N ≥1 is

N

i

iN

xN

xxW1

1 .0,0),exp(),,(

.)(),,( 11N N

i iiNN xSxxW

6 A simple application in China

• Does the Chinese Reform and Open Policy practice generate a fair income distribution?

• How much had Chinese people been satisfied by the great increase in national income in the past several decades?

9872.,4739. 2

)3392.65( RBeijingHunan

,)(2126.3392. 1994)1775.14()9977.31(

tDUrbanUrbanUrbanRural

1994;0

1994;1,9869.2

t

tDR t

• Figure 1 Impartial regional income distribution

1980 1990 2000 20100

5

10

15

20

25

30

35

40

45

50

year

avera

ge w

age(1

03 y

uan)

Beijing

Hunan

0 10 20 300

5

10

15

20

25

30

35

40

45

50

wage in Hunan(103 yuan)

wage in B

eijin

g(1

03 y

uan)

• Figure 2 Unfair income distributions between urban and rural

1980 1990 2000 20100

5

10

15

20

25

year

incom

e(1

03 y

uan)

urban

rural

0 2 4 60

5

10

15

20

25

rural income(103 yuan)

urb

an incom

e(1

03 y

uan)

• Inspired by supported evidences for that more income brings greater satisfaction among income groups at a point in life and a cohort’s satisfaction remains constant throughout the life span (Easterlin, 2001), we thus construct the following SF and SSI:

NixNx

xSN

nntt

it

titit ,,2,1,

1),exp()(

12

)1

exp(1

2

N

i it

t

xNW

N

i it

Nttt

xNxx

W1

21 1),min(

exp(

• Table SSI in ChinaⅡyear China Beijing Hunan Max Min year China Beijing Hunan Max Min

1981 .4291 .4768 .3727 .6593 .3170 1995 .4215 .7098 .3723 .7679 .2644

1982 .4253 .4708 .3689 .7193 .3214 1996 .4229 .7343 .3363 .7941 .2565

1983 .4292 .5021 .3747 .7360 .3190 1997 .4298 .7707 .3279 .7848 .2662

1984 .4293 .4936 .3746 .7532 .3201 1998 .4353 .7713 .3921 .8038 .2493

1985 .4404 .5766 .4055 .7708 .3051 1999 .4284 .7739 .3837 .8329 .2526

1986 .4327 .5517 .3961 .7823 .3139 2000 .4183 .7847 .3748 .8280 .2581

1987 .4417 .5740 .4344 .7712 .3010 2001 .4243 .7911 .3952 .8343 .2530

1988 .4386 .5728 .4304 .7333 .2912 2002 .4175 .7877 .3877 .8304 .2582

1989 .4378 .5681 .4182 .7027 .2897 2003 .4104 .7963 .3763 .8222 .2591

1990 .4396 .5911 .4103 .6933 .2901 2004 .4113 .8059 .3756 .8193 .2588

1991 .4435 .5941 .4027 .6850 .2807 2005 .4083 .8065 .3587 .8081 .2614

1992 .4343 .6038 .4004 .7263 .2841 2006 .4121 .8159 .3579 .8245 .2600

1993 .4437 .6611 .4263 .7679 .2592 2007 .4215 .8089 .3718 .8281 .2579

1994 .4446 .6977 .4028 .7563 .2606

• Figure 3 Inequality of SSI

1980 1985 1990 1995 2000 2005 20100.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

year

SS

I

China

Beijing

Hunan

Max

Min

7 Conclusions• First, the uniqueness of homogenous and symmetrically

differentiable SSF shows that economists might be more unified about the analytic form of SWF.

• Second, the concept of social satisfaction is similar to individual satisfaction as well as the social welfare and the individual utility.

• Another fact is that the equality on welfare does not necessarily mean the equality on income distribution.

• In addition, the evidence from China shows that the social welfare may not increase even if the social income level increases a lot.

• Finally, an important but unresolved question is that whether the impartial income distribution can lead to an equal distribution of income or satisfaction. After all, the fairness concept should be about both impartiality and equality.

• Appendices• A Proof of theorem 2.3• B Proof of theorem 5.1• C Proof of theorem 5.3• D Original datasets

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